Abstract
This paper deals with the critical parameter equations for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the selfsimilar supersolution and subsolution, we obtain the critical global existence parameter equation. The critical Fujita type is conjectured with the aid of some new results.
Mathematics Subject Classification (2000). 35K55; 35K57.
Keywords:
degenerate parabolic system; global existence; blowup1 Introduction
In this paper, we consider the following degenerate parabolic equations
coupled via nonlinear boundary flux
with continuous, nonnegative initial data
compactly supported in ℝ_{+}, where p_{i }> 1, q_{i }> 0, (i = 1, 2, ..., k) are parameters.
Parabolic systems like (1.1)(1.3) appear in several branches of applied mathematics. They have been used to models, for example, chemical reactions, heat transfer, or population dynamics (see [1] and the references therein).
As we shall see, under certain conditions the solutions of this problem can become unbounded in a finite time. This phenomenon is known as blowup, and has been observed for several scalar equations since the pioneering work of Fujita [2]. For further references, see the review by Leivine [3]. Blowup may also happen for systems (see [47]). Our main interest here will be to determine under which conditions there are solutions of (1.1)(1.3) that blow up and, in the blowup case, the speed at which blowup takes place, and the localization of blowup points in terms of the parameters p_{i}, q_{i}, (i = 1, 2, ..., k).
As a precedent, we have the work of Galaktionov and Levine [8], where they studied the single equation
It was shown if 0 < q_{2 }≤ q_{0 }= (p_{1 }+1)/2, then all nonnegative solutions of (1.4) are global in time, while for q_{2 }> q_{0 }there are solutions with finite time blowup. That is, q_{0 }is the critical global existence exponent. Moreover, it was shown that q_{c }:= p_{1 }+ 1 is a critical exponent of Fujita type. Precisely, q_{c }has the following properties: if q_{0 }< q_{2 }≤ q_{c}, the all nontrivial nonnegative solutions blow up in a finite time, while global nontrivial nonnegative solutions exist if q_{2 }> q_{c}.
We remark that there are some related works on the critical exponents for (1.1)(1.3) in special cases.
In [911], the authors consider the case for p_{i }= 1, (i = 1, 2, ..., k).
In [12], the authors consider the case for k = 2.
For the system (1.1)(1.3), instead of critical exponents there are critical parameter equations, one for global existence and another of Fujita type. This is the content of our first theorem.
To state our results, we introduce some useful symbols. Denote by
A series of standard computations yield
We shall see that det A = 0 is the critical global existence parameter equation. Let (α_{1}, α_{2}, ..., α_{k})^{T }be the solution of the following linear algebraic system
that is
We define
Theorem 1.1.
(I) If (i.e. det A ≥ 0), every nonnegative solution of (1.1)(1.3) is global in time.
(II) If (i.e. det A < 0) and there exists j (1 ≤ j ≤ k) such that α_{j }+ β_{j }≤ 0, then every nonnegative, nontrivial solution blows up in finite time.
(III) If (i.e. det A < 0), with α_{i }+ β_{i }> 0 (i = 1, 2, ...,k), there exist nonnegative solutions with blowup and nonnegative solutions that are global.
Therefore, the critical global existence parameter equation is
and the critical Fujita type parameter equation is
The values of α_{i}, β_{i }(i = 1, 2, ..., k) are the exponents of selfsimilar solutions to problem (1.1)(1.2). Such selfsimilar solutions are studied in Section 2, and play an important role in the proof of Theorem 1.1.
Let us observe that if we take k = 2, the critical parameter equations coincide with those found in [12].
The rest of this paper is organized as follows. In the next section, we study the existence of selfsimilar solutions of different type. In Section 3 we give some results concerning existence, comparison, monotonicity and uniqueness. In Section 4 we find the critical parameter equations (Theorem 1.1).
2 Selfsimilar solutions
In this section, we consider different kinds of selfsimilar solutions of problem (1.1)(1.2). We have the following results.
Theorem 2.1. Let
If
there is a selfsimilar solution of problem (1.1)(1.2) blowing up in a finite time T > 0, of form (2.1). Moreover, the support of f_{i }is ℝ_{+ }if β_{i }> 0, and a compact set if β_{i }≤ 0 (i = 1, 2, ..., k).
Theorem 2.2. Let
(a) If
then there exist functions f_{i }positive in ℝ_{+}, such that u_{i }given in (2.3) is a selfsimilar solution of problem (1.1)(1.2) global in time. These solutions have α_{i }> 0 and thus their initial data are identically zero. Then β_{i }< 0 (i = 1, 2, ...,k).
(b)If
then there exist functions f_{i}, compactly supported in ℝ_{+}, such that u_{i }given in (2.3) is a selfsimilar solution of problem (1.1)(1.2) global in time. These solutions have α_{i }< 0 and thus they decay to zero as t → ∞. Then β_{i }> 0, and hence their supports expand as time increases.
Remark 2.2. If there exists j (1 ≤ j ≤ k) such that α _{j }+ β_{j }≤ 0, there are no profiles f_{i }∈ L^{1}(ℝ_{+}) such that u_{i }(i = 1, 2, ..., k,) given by (2.3) is a solution. Indeed
Then, if α_{j }+ β_{j }≤ 0, the mass of u_{j }would not increase, a contradiction.
Theorem 2.3. Let
If
for any α_{1 }> 0, there is a selfsimilar solution of problem (1.1)(1.2) global in time of form (2.5) where
Moreover, the supports of f_{i }(i = 1, 2, ..., k) are compact.
Remark 2.3. The solutions are in principle weak. However, if they are positive everywhere, they are also classical.
In order to prove these theorems, we will use the following results of Gilding and Peletier (see [1315]):
Theorem 2.4. Let a, b, V ∈ ℝ and U ≥ 0. For fixed a and b, let S_{A }denote the set of values of (U, V) such that there exists a weak, nonnegative, compactly supported solution f_{1 }of
and let S _{B }denote the set of values (U, V) for which there exists a bounded, positive, classical solution f_{1 }of (2.8)(2.10).
(a) If b < 0 and 2a + b < 0, then S _{A }= {(0, 0)} and S_{B }= Ø.
(b) If b < 0 and 2a + b = 0, then S _{A }= {(0, V): 0 ≤ V < ∞} and S _{B }= Ø.
(c) If b ≤ 0 and 2a + b > 0, then there exists a unique V_{* }such that and S _{B }= {(U, V): 0 ≤ U < ∞, , where V_{* }> 0 if a + b < 0, V_{* }= 0 if a + b = 0, and V_{* }< 0 if a + b > 0.
(d) If b > 0 and a ≥ 0, then there exists a unique V_{* }< 0 such that and S _{B }= Ø.
(e) If b > 0 and a < 0, or b = 0 and a ≤ 0, then S _{A }= {(0, 0)} and there exists a unique V_{* }such that S _{B }= {(U, U^{(p1+1)/2}V_{*}): 0 ≤ U < ∞}, where V_{* }< 0 if b > 0 and V_{* }= 0 if b = 0.
Moreover, for each (U, V) ∈ S _{A }∪ S _{B }there exists at most one weak solution of (2.8)(2.10).
Remark 2.4. In the case where a = ((p_{1 } 1)/2)b > 0, we have V_{* }= 1. This is a consequence of the existence for a selfsimilar solution of exponential form for the scalar problem (1.4) with q_{2 }= (p_{1 }+ 1)/2 (see [8]).
Proof of Theorem 2.1. We consider solutions of form (2.1). Imposing that the porous equations (1.1) are fulfilled, we get the following relations for the parameters:
On the other hand, the boundary conditions (1.2) imply that
Solving the linear systems (2.11)(2.12), we get that α_{i}, β_{i }(i = 1, 2, ..., k) are given by (1.5) and (1.6). Therefore, α_{i }< 0 (i = 1, 2, ..., k) if and only if . On the other hand, the profiles must satisfy
plus the boundary conditions
Then f_{i }satisfy (2.8) with coefficients a_{i }= β_{I}, b_{i }= α_{i }(i = 1, 2, ..., k). Thus, Theorem 2.4 parts (d) and (e) says that there is an oneparameter family (parameter U_{i}) of (2.8) satisfying
where V_{*i }< 0 (i = 1, 2, ..., k) are constants. The profile f_{i }has compact support if β_{i }≤ 0 and is positive in ℝ_{+ }if β_{i }> 0. We choose U_{i }such that the boundary conditions (2.14) are fulfilled, that is
Taking logarithms, this is equivalent to
As (i.e. det A ≠ 0), the above system has a unique solution. □
Proof of Theorem 2.2. We are considering solutions of the form (2.3). Imposing that the equations (1.1) and that boundary conditions (1.2) are fulfilled, we get that the exponents should satisfy the relations (2.11)(2.12). Hence they are given by (1.5)(1.6). Moreover, the boundary conditions for the profiles are given by (2.14). However, the equations for the profiles are now different:
Thus, f_{i }satisfy (2.8) with coefficients a_{i }= β_{i}, b_{i }= α_{i }(i = 1, 2, ..., k).
(I) If α_{i }> 0, that is, if (2.4) holds, then β_{i }< 0 (i = 1, 2, ..., k). Therefore, applying Theorem 2.4 part (d) as in the proof of Theorem 2.1, and taking the solutions of (2.15) as values for parameters, we obtain that there exist positive profiles f_{i }(i = 1, 2, ..., k) solving (2.16) and satisfying (2.14).
(II) If α_{i }< 0 and α_{i }+ β_{i }> 0 (i = 1, 2, ..., k), we can apply Theorem 2.4 part (c) as in the proof of Theorem 2.1 and taking the solutions of (2.15) as the parameters, we obtain that there exist compactly supports profiles f_{i }(i = 1, 2, ..., k) solving (2.16) and satisfying the boundary conditions (2.14).
Proof of Theorem 2.3. We are considering solutions of the form (2.5). Though the boundary conditions (1.2) impose (2.12) again, now equations (1.1) impose different relations for the exponents. Namely
Thus,
There are nontrivial solutions of (2.18) if and only if (i.e. det A = 0). In this case, β_{1}, α_{I}, β_{i }(i = 2, ...,k) are related to α_{1 }by (2.7).
The boundary conditions for the profiles are again given by (2.14), while the equations for the profiles are given by (2.16). If α_{1 }> 0, then β_{1}, α_{i}, β_{i }> 0 (i = 2, ..., k) and β_{i }= ((p_{i } 1)/2)α_{i }(i = 1, ..., k). Hence, using Remark 2.4, we have solutions of (2.16) with V_{*i }= 1 (i = 1, 2, ...,k). Choosing one of the solutions of (2.15) with righthand side zero (again we are using (i.e. det A = 0)), we obtain that there exist compactly supported profiles f_{i }(i = 1, 2, ..., k) solving (2.16) and satisfying (2.14).
3 Existence and uniqueness
First, we state a theorem that guarantees the existence of a solution. It can be obtained using a standard monotonicity argument following ideas from [16].
Theorem 3.1. Given continuous, compactly supported initial data u_{0i}(x) (i = 2, ..., k), there exists a local in time continuous weak solution of (1.1)(1.3). Moreover, if the initial data are smooth and compatible in sense that
then the solution has continuous time derivatives down to t = 0.
Proof. Let us consider the Neumann problem
with r > 1. We define the operator as , where w(x, t) is the unique solution of (3.1) with r = p_{i }and initial condition w_{0}(x) = u_{0i}(x) .
It has been proved in [17] that is continuous and compact. Moreover, they are order preserving.
Now let . Using the method of monotone iterations, one can prove that there exist τ > 0 such that A has a fixed point in C([0, τ]). This fixed point provides us with a continuous weak solution of (1.1)(1.3) up to time τ.
In order to obtain the regularity of the solution with compatible initial data, we only have to observe that the solution of (3.1) is regular if (see [18]).
Remark 3.1. If the initial data are compactly support, the solution u_{i }(i = 1, 2, ..., k) also has compact support as long as it exists.
Remark 3.2. If the initial data are nontrivial, we can assume that they satisfy u_{0i}(x) > 0 (i = 1, 2, ..., k). If not, u_{i}(0, t) (i = 1, 2, ..., k) eventually become positive (compare with a Barenblatt solution of the corresponding equation).
Next, we define what called a subsolution and a supersolution for (1.1)(1.2).
Definition 3.1. is a subsolution of (1.1)(1.2) if it satisfies
Definition 3.2. We call a supersolution of (1.1)(1.2) of it satisfies (3.2)(3.3) with the opposite inequalities.
With these definitions of super and subsolutions, we can state a comparison lemma.
Lemma 3.1 Let be a supersolution and be a subsolution. If
with
then
as long as both super and subsolutions exist.
Proof. It is standard, therefore we omit the details. Assume that the result is false. Let t_{0 }be the maximum time such that
up to t_{0}. This time t_{0 }must be positive, by continuity. At that time, we must have for some j (1 ≤ j ≤ k). Let us assume that . Now the result follows by an application of Hopf's lemma. Indeed, satisfies a uniformly parabolic equation in a neighborhood of x = 0, attains a minimum at (0, t_{0}), and the corresponding flux is greater or equal than zero, a contradiction.
Now we state a lemma that guarantees that, for certain initial data, the solution of (1.1)(1.3) increases in time.
Lemma 3.2 Let u_{0i}(x) be the initial data for (1.1) (1.3) such that u_{0i}(x) are smooth, satisfy the compatibility condition at the boundary and . Then u_{i}(x, t) increases in time, i.e., u_{it}(x, t) ≥ 0 (i = 1, 2, ...,k).
Proof. Let w_{i }= u_{it}. Then, as the solutions are smooth (Theorem 3.1), we can differentiate to obtain the (w_{1}, ..., w_{k}) is a solution of
with initial data satisfying
To conclude the proof we apply the maximum principle. Due to the degeneration of the equations this cannot be done directly. A standard regularization procedure is needed (see [8] for details).
Next, we deal with the problem of uniqueness versus nonuniqueness for (1.1)(1.3) on the case of vanishing initial data (u_{0i}(x) = 0, i = 1, 2, ..., k).
Theorem 3.2
(a) Let . Then there exists a nontrivial solution with zero initial data that becomes positive at × = 0 instantaneously. Then there is no uniqueness for problem (1.1)(1.3) with zero initial data.
(b) Let . Then the solution of (1.1)(1.3) with zero initial data is unique.
Proof.
(a) The selfsimilar solutions constructed in Theorem 2.2 become positive at x = 0 instantaneously.
(b) We can construct small supersolution with the aid of the selfsimilar ones of exponential form that we found in Theorem 2.3. First, choose such that .
where α_{1 }> 0 is arbitrary and β_{1}, α_{i}, β_{i}, (i = 2, ..., k) are given by (2.7). Now we observe that be a supersolution is a supersolution of (1.1)(1.3) as long as . By the comparison Lemma 3.1, we obtain that every solution has initial data identically zero satisfies
As can be chosen as small as we want (using τ negative and large enough) we conclude that .
4 Blowup versus global existence
We devote this section to prove Theorem 1.1. We borrow ideas from [8]. However, the fact that we are dealing with a system instead of a single equation forces us to develop a significantly different proof. We will organize the proof in several lemmas.
Our first lemma proves part (I) of Theorem 1.1.
Lemma 4.1. If (i.e. det A ≥ 0), every nonnegative solution of (1.1)(1.3) is global in time.
Proof. It is enough to construct global supersolutions with initial data as large as needed. We achieve this with the aid of the selfsimilar solutions of exponential form that we found in Theorem 2.3.
First we choose such that and we let
where α_{1 }> 0 is arbitrary and β_{1}, α_{i}, β_{i}, (i = 2, ..., k) are given by (2.7). Now we observe that is a supersolution of (1.1)(1.3) as long as . This can be done by choosing τ large enough. This also allows to assume . Then, by the comparison Lemma 3.1, we obtain that every solution is global.
Now we construct subsolutions with finite time blowup.
Lemma 4.2. Let (i.e. det A < 0), then there exist compactly supported functions g_{i }(i = 1, 2, ..., k), such that
is a subsolution of (1.1)(1.2).
Proof. To satisfy (3.2) and (3.3), we need that
We choose
Inserting this in the equation, we get
Hence, it is enough to impose
that is
The boundary conditions impose
Let
Then conditions (4.2) become
We fix b_{i }= 1 (i = 1, 2, ⋯, k) and then A_{i }large enough (and thus a_{i }small) to satisfy (4.1) and (4.3).
Corollary 4.1 Let (i.e. det A < 0). Then there exist solutions of (1.1)(1.3) that blow up in a finite time.
Proof. We only have to apply Lemma 3.1, to obtain that every solution (u_{1}, ⋯, u_{k}) that begins above the subsolutions provided by Lemma 4.2 has finite time blowup.
Lemma 4.3 Let (i.e. det A < 0). If there exists j (1 ≤ j ≤ k) such that α_{j }+ β_{j }≤ 0, then every nontrivial solution of (1.1)(1.3) blows up in finite time.
Proof. Without loss of generality, we consider the case α_{1 }+ β_{1 }≤ 0.
Assume that there exists a global nonnegative nontrivial solution of (1.1)(1.3), we make the following change of variables
These functions satisfy
As u_{i}(x, t) (i = 1, 2, ..., k) are by hypothesis global, the same is true for φ_{i }(i = 1, 2, ..., k,). We will construct a solution to system (4.5)(4.6) increasing with time, with initial data such that . We will prove that cannot exists globally, thus contradicting the global existence of (u_{1}, ⋯, u_{k}). In order to achieve our goal, we use an adaptation for systems of the general monotonicity for single quasilinear equation described in [19].
We take initial data satisfying
and the compatibility conditions
Hence, arguing as in Lemma 3.2, we have that .
Following an idea for scalar equation from [8], we set
where h is the Barenblatt profile
Then we have
The last expression is nonnegative if β_{1 } 1/(p_{1 }+ 1) ≤ 0 and α_{1 } 1/(p_{1 }+ 1) ≥ 0. But these two conditions are equivalent α_{1 }+ β_{1 }≤ 0.
Now we take such that . We take as a solution to
There is oneparameter family of solution to this equation (see Theorem 2.4), with . Hence,
Moreover,
where V_{*i }< 0 is a constant and U_{i }is the free parameter.
We still have to control the boundary conditions. In order to do this, we choose the constants c, b and U_{i }(i = 2, ...,k) conveniently. They have to satisfy
Thus, we choose
where c_{i }(i = 2, ..., k) and γ are positive constants. Taking b small enough, the initial data is below (u_{1}(ξ_{1},0), ...,u_{k}(ξ_{k}, 0)). This can be done as u_{0i }(i = 1, 2, ... k) can be assumed to be positive at the origin.
To conclude the proof, we will show that converge to a selfsimilar profile that does not exist in this range of parameters.
Lemma 4.4. There exists j (1 ≤ j ≤ k) such that
Proof. It is clear that . Let us suppose that
In the original variables , we have that for any M > 0 there is a value such that
Now we will check that, under these conditions, we can put one of the blowing up subsolutions constructed in Lemma 4.2 below these data. This would lead to a contradiction, as is global. In order to do this, we need
The first equation says that the height at x = 0 of is bigger than that of , and the second says that the support of is bigger than the support of . Imposing analogous conditions for and we get
Taking T = 1 + t_{0}, then a_{i }small enough and A_{i }large enough (i = 1, 2, ..., k), and then M large, then the 2k conditions (4.9)(4.10) are fulfilled.
Let us remark this parametric evolution comparison method to prove global nonexistence for arbitrary data first introduced in [20], for scalar quasilinear heat equation.
End of the proof of Lemma 4.3. Let us assume that (4.7) holds. Using standard arguments, see [8], we may pass to the limit to obtain that
Hence, in (0, ξ_{10}), z ≥ c > 0,
We conclude that z and therefore cannot be unbounded at ξ_{1 }= 0. In particular, Then, considering the regularity of in the region where , we can pass to the limit in the boundary condition for to obtain that
However, as α_{1 }+ β_{1 }≤ 0, problem (4.11)(4.12) does not have a nontrivial solution, see Theorem 2.4.
If (4.7) holds for some j > 1, we can proceed as before to obtain that . Thus, we can pass to the limit in the boundary condition for , obtaining
As , this implies that is finite for every ξ_{j+1 }≥ 0. We get the same contradiction as before.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
We would like to thank Professor Dimitru Motreanu, Christopher Rualizo and the referees for their valuable comments and suggestions.
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